Math for the Building Trades — ANEW Pre-Apprenticeship Program
Vocab Key Terms
Before working through the formulas, know these terms. Tap any card to see the example.
Tap a card to expand the example.
1 Common Units & the Tape Measure
In the building trades, most measurements come from a tape measure. The tape is divided into feet, inches, and fractions of inches. Every 12 inches the tape marks a new foot.
On the job: Tape measures in the US read feet and inches. The smallest marks are 1/16 inch. The red diamonds every 19.2 inches are for truss layout. Always measure twice, cut once.
Interactive Tape Measure
Adding and Subtracting Measurements
Handle feet and inches in separate columns — just like adding two-column numbers. If inches total 12 or more, carry one foot. If subtracting more inches than you have, borrow one foot (= 12 inches) from the feet column.
Add and Subtract Feet and Inches
First
ftin
Second
ftin
Rule: Feet stay with feet, inches stay with inches. Never mix units in the same column.
2 Perimeter
A shape's perimeter is the total distance around all of its edges. On a job site, perimeter tells you how much fencing, baseboard, trim, or framing material to order. It is always measured in linear units (ft, in, m).
On the job: Ordering baseboard for a room means measuring the perimeter. Fencing a yard, running conduit around a wall, strapping a load — all perimeter problems.
Worked Example — Rectangle
A room is 14 ft long and 10 ft wide. What is the perimeter?
Step 1: Write the formula: P = 2(L + W)
Step 2: Substitute values: P = 2(14 + 10)
Step 3: Add inside the parentheses: P = 2(24)
Step 4: Multiply: P = 48 ft
P = 48 ft
Worked Example — Triangle
A triangular brace has sides 3 ft, 4 ft, and 5 ft. What is the perimeter?
Step 1: Write the formula: P = a + b + c
Step 2: Substitute: P = 3 + 4 + 5
Step 3: Add: P = 12 ft
P = 12 ft
Try It
Square — P = 4s
How to solve▶
Rectangle — P = 2(l+w)
How to solve▶
Triangle — P = a+b+c
How to solve▶
Square
P = 4s
Rectangle
P = 2(l+w)
Triangle
P = a+b+c
Any polygon
P = all sides
3 The Pythagorean Theorem
A right triangle has exactly one 90° angle. The side opposite that angle is the hypotenuse (c) — always the longest side. The other two sides are the legs (a and b). The Pythagorean Theorem states: a² + b² = c².
Common mistake: √(a² + b²) does NOT equal a + b. Square each leg first, then add, then take the square root. Never skip the order of operations.
Worked Example 1 — Find the Hypotenuse (3-4-5)
A right triangle has legs a = 3 ft and b = 4 ft. Find hypotenuse c.
Step 1: Write the formula: c = √(a² + b²)
Step 2: Square each leg: 3² = 9, 4² = 16
Step 3: Add: 9 + 16 = 25
Step 4: Square root: √25 = 5
c = 5 ft
Worked Example 2 — Find a Leg (10-8-6)
A right triangle has hypotenuse c = 10 ft and leg b = 8 ft. Find leg a.
Step 1: Rearrange the formula: a = √(c² − b²)
Step 2: Square the known sides: 10² = 100, 8² = 64
Step 3: Subtract: 100 − 64 = 36
Step 4: Square root: √36 = 6
a = 6 ft
Try It
Pythagorean Theorem — Solve for Any Side
On the job: The 3-4-5 rule (and multiples: 6-8-10, 9-12-15) checks square corners on slabs, walls, and decks. If the diagonal does not match c, the corner is not square.
4 Converting Between Units
The conversion factor must be applied once for every dimension. Length is 1D — multiply or divide by the factor once. Area is 2D — apply it twice (which is squaring it). Volume is 3D — apply it three times (cubing it).
From
To
Operation
Factor
Why
in
ft
÷ 12
12
1D — 12¹
ft
in
× 12
12
1D — 12¹
in²
ft²
÷ 144
144
2D — 12²
ft²
in²
× 144
144
2D — 12²
in³
ft³
÷ 1,728
1,728
3D — 12³
ft³
in³
× 1,728
1,728
3D — 12³
The interactive below proves why. Step through the Area tab to see why in² needs ÷144. Then the Volume tab for ÷1,728. Use the Calculator tab to convert any value with the math shown step by step.
2D — Area
Square inches to square feet
1 ft = 12 in. So why divide by 144 — not 12 — to convert in² to ft²? Step through to see inside the grid.
1 / 7
The ruleRaise the conversion factor to the power of the dimension. Length: 12¹. Area: 12²=144. Volume: 12³=1,728.
Wrong144 in² ÷ 12 = 12Mixed units: 1 ft × 12 in ≠ 1 ft²
Now we add the z-axis. Each direction needs its own ÷12. Watch the cube fill in layer by layer.
1 / 7
The takeawayx, y, z — each direction needs its own conversion. 12×12×12=1,728.
Wrong1,728 in³ ÷ 12 = 144 (only x converted)
Correct1,728 in³ ÷ 1,728 = 1 ft³
Live Calculator
Convert any measurement
Enter a value, pick a conversion, see the step-by-step math.
1D
÷ 12
in → ft
2D
÷ 144
in² → ft²
3D
÷ 1,728
in³ → ft³
Unit Conversion — Step by Step
Coming next — Unit 5: Volume. Add a third direction (the z-axis) and area becomes volume. It is the same pattern, just one more power: length = 12, area = 12² = 144, volume = 12³ = 1,728. You will build it out in Unit 5.
5 Area
A shape's area is the two-dimensional space it occupies, measured in square units (ft², in², m²). On a job site, area tells you how much flooring, roofing, concrete, paint, or siding to order.
On the job: Flooring is sold by the square foot. Roofing by the square (100 ft²). Concrete is ordered by the cubic yard — but you calculate the slab area first to get the volume. Always add 10% for waste.
Worked Example — Rectangle (from the notes)
A rectangle is 7 units wide and 4 units tall. What is the area?
Step 1: Write the formula: A = L × W
Step 2: Substitute: A = 7 × 4
Step 3: Multiply: A = 28 units²
Think about it: 7 columns of 4 unit squares each = 28 squares total.
A = 28 units²
Try It — All Shapes
Square — A = s²
How to solve▶
Rectangle — A = l×w
How to solve▶
Triangle — A = ½bh
How to solve▶
Parallelogram — A = bh
How to solve▶
Trapezoid — A = ½(b₁+b₂)h
How to solve▶
Circle — A = πr²
How to solve▶
Rectangle
A = l × w
Triangle
A = ½bh
Parallelogram
A = bh
Trapezoid
A = ½(b₁+b₂)h
Circle Area
A = πr²
Circumference
C = 2πr
Always use perpendicular height — the straight vertical distance, not the slanted side. If you are given the diameter of a circle, divide by 2 to get the radius first.
6 Composite Figures
A composite figure is a shape made from two or more simple shapes joined together. To find the area, you decompose it — break it apart into rectangles, triangles, or other shapes you already know — find each area, then add or subtract them.
On the job: Room layouts are almost never pure rectangles. An L-shaped room, a floor with a closet cut out, a deck with a notch — all composite figures. Same method every time: break it into pieces, find each piece, add them up.
Worked Example — L-Shape
An L-shaped room: total width 10 ft, total height 8 ft, with a 4 ft × 3 ft rectangle cut from the top-right corner.
Step 1: Break it into two rectangles. Rectangle A is 10 ft × 5 ft (the bottom). Rectangle B is 6 ft × 3 ft (the top-left).
Step 2: Find area of A: 10 × 5 = 50 ft²
Step 3: Find area of B: 6 × 3 = 18 ft²
Step 4: Add: 50 + 18 = 68 ft²
Total area = 68 ft²
Try It — L-Shape
Enter the outer dimensions and cutout size. The diagram updates to show how it breaks into two rectangles.
Outer Box
Cutout (top-right)
How to solve▶
Two strategies: (1) Add — split the shape into parts and add their areas. (2) Subtract — start with the big outer rectangle, subtract the cutout. Both give the same answer.
Ready to Practice?
The Worksheet has 136 problems covering every shape. Each problem has a labeled drawing, step-by-step how-to, and a graph view.